Assertion (A) : PA and PB are triangles to a circle with centre O such that `angleAOB=110^(@)`, then `angleAPB=90^(@)`
Reason (R ) : The length of two tangents drawn from an external point are equal .

Options

A. Both A and R are true and R is the correct explanation for A.
B. Both A and R are true and R is not correct explanation for A.
C. A is true but R is false.
D. A is false but R is true.

Answer

To solve the problem, we need to analyze the assertion and the reason given:

### Step-by-Step Solution:

1. **Understanding the Assertion**:
- The assertion states that if PA and PB are tangents to a circle with center O, and the angle AOB is 110°, then the angle APB is 90°.

2. **Drawing the Diagram**:
- Draw a circle with center O. Mark point P outside the circle. Draw tangents PA and PB from point P to the circle, touching the circle at points A and B respectively.

3. **Identifying Angles**:
- According to the properties of tangents, the angle between the tangent and the radius at the point of contact is 90°. Thus, ∠OAP = 90° and ∠OBP = 90°.

4. **Using the Angle Sum Property**:
- In quadrilateral OAPB, the sum of the angles is 360°. We know:
- ∠OAP = 90°
- ∠OBP = 90°
- ∠AOB = 110°
- Let ∠APB = x. Therefore, we can write the equation:
\[ x + 90° + 110° + 90° = 360° \]

5. **Simplifying the Equation**:
- Combine the known angles:
\[ x + 290° = 360° \]
- Now, isolate x:
\[ x = 360° - 290° = 70° \]

6. **Conclusion about Assertion**:
- We found that ∠APB = 70°, not 90°. Hence, the assertion is false.

7. **Understanding the Reason**:
- The reason states that the lengths of two tangents drawn from an external point are equal. This is a true statement based on the properties of tangents.

8. **Final Evaluation**:
- Since the assertion is false and the reason is true, we conclude that the assertion (A) is false and the reason (R) is true.

### Final Answer:
- Assertion (A) is false.
- Reason (R) is true.